Deflection of Indeterminate Structure Session Matakuliah: S0725 – Analisa Struktur Tahun: 2009

Bina Nusantara University 3 Contents Elastic Beam Theory Moment area Conjugate Beam Method

Bina Nusantara University 4 Introduction What is deflection ? Deflection can occur from various causes, such as loads, settlement, temperature or fabrication error of material. Deflection must be limited in order to prevent cracking and damaging of structure.

Bina Nusantara University 5 Introduction What is caused of structure deflection ? In general It caused by its internal loading such as Normal force, shear force or bending moment

Bina Nusantara University 6 Introduction What is caused of Beam & Frames deflection? It caused by internal bending What is caused of truss deflection ? It caused by internal axial forces

Bina Nusantara University 7 Introduction Deflection Diagram represent the Elastic Curve for the points at the centroids of cross-sectional area along the members

Bina Nusantara University 8 Introduction Deflection on supports : (1) Roller   = 0 A

Bina Nusantara University 9 Introduction Deflection on supports : (2) Pin   = 0 A

Bina Nusantara University 10 Introduction Deflection on supports : (3) Fixed Support   = 0 ;  = 0

Bina Nusantara University 11 Introduction Deflection on supports : (4) Fixed Connected Joint  causes the joint to rotate the members by the same amount of   

Bina Nusantara University 12 Introduction Deflection on supports : (5) Pin Connected Joint  the members will have a different slope or rotation at pin, since the pin can’t support moment 11 22

Bina Nusantara University 13 Introduction Sign Convention  Bending Moment  M+ Longitudinal axis M+

Bina Nusantara University 14 Introduction Sign Convention  Bending Moment  M- Longitudinal axis M-

Bina Nusantara University 15 Introduction Deflection curve : B A P2 P1 Moment - + B A Deflection curve

Bina Nusantara University 16 x P P y Elastic curve The deflection is measured from the original neutral axis to the neutral axis of the deformed beam. The displacement y is defined as the deflection of the beam. It may be necessary to determine the deflection y for every value of x along the beam. This relation may be written in the form of an equation which is frequently called the equation of the deflection curve (or elastic curve) of the beam Importance of Beam Deflections A designer should be able to determine deflections, i.e. In building codes y max <=L beam /300 Analyzing statically indeterminate beams involve the use of various deformation relationships.

Bina Nusantara University 17 Elastic Beam Theory x P P y Elastic curve The deflection is measured from the original neutral axis to the neutral axis of the deformed beam. The displacement y is defined as the deflection of the beam. It may be necessary to determine the deflection y for every value of x along the beam. This relation may be written in the form of an equation which is frequently called the equation of the deflection curve (or elastic curve) of the beam

Bina Nusantara University 18 Elastic Beam Theory Importance of Beam Deflections A designer should be able to determine deflections, i.e. y max <=L beam /300 Analyzing statically indeterminate beams involve the use of various deformation relationships.

Bina Nusantara University 19 Double-Integration Method The deflection curve of the bent beam is In order to obtain y, above equation needs to be integrated twice. y  Radius of curvature y x

Bina Nusantara University 20 Double-Integration Method An expression for the curvature at any point along the curve representing the deformed beam is readily available from differential calculus. The exact formula for the curvature is

Bina Nusantara University 21 The Integration Procedure Integrating once yields to slope dy/dx at any point in the beam. Integrating twice yields to deflection y for any value of x. The bending moment M must be expressed as a function of the coordinate x before the integration Differential equation is 2 nd order, the solution must contain two constants of integration. They must be evaluated at known deflection and slope points (i.e. at a simple support deflection is zero, at a built in support both slope and deflection are zero) Double-Integration Method

Bina Nusantara University 22 Sign Convention Positive Bending Negative Bending Assumptions and Limitations  Deflections caused by shearing action negligibly small compared to bending  Deflections are small compared to the cross-sectional dimensions of the beam  All portions of the beam are acting in the elastic range  Beam is straight prior to the application of loads Double-Integration Method

Bina Nusantara University 23 First Moment –Area Theorem The first moment are theorem states that: The angle between the tangents at A and B is equal to the area of the bending moment diagram between these two points, divided by the product EI. Moment Area

Bina Nusantara University 24 Tangent at A  A B  Tangent at B  dd dd x dx ds M Moment Area

Bina Nusantara University 25 The second moment area theorem states that: The vertical distance of point B on a deflection curve from the tangent drawn to the curve at A is equal to the moment with respect to the vertical through B of the area of the bending diagram between A and B, divided by the product EI. Moment Area Second Moment –Area Theorem

Bina Nusantara University 26 Procedure 1.The reactions of the beam are determined 2.An approximate deflection curve is drawn. This curve must be consistent with the known conditions at the supports, such as zero slope or zero deflection 3.The bending moment diagram is drawn for the beam. Construct M/EI diagram 4.Convenient points A and B are selected and a tangent is drawn to the assumed deflection curve at one of these points, say A 5.The deflection of point B from the tangent at A is then calculated by the second moment area theorem Moment Area

Bina Nusantara University 27 P PL L P A B  Tangent at A Tangent at B  Moment Area Problem

Bina Nusantara University 28 P PL L P A B  Tangent at A Tangent at B  PL M Moment Area

Bina Nusantara University 29 WL Tangent A L A W N per unit length B  = ? Moment Area

Bina Nusantara University 30 Example L a P a P P P a Pa Tangent A A  = ? Moment Area

Bina Nusantara University 31 Moment Area

Bina Nusantara University 32 Conjugate Beam The method requires the same amount of computation as the moment-area theorems to determine a beam’s slope or deflection. However, the method relies only on the principles of statics, its application will be more familiar Dr Yan Zhuge lecturer notes

Bina Nusantara University 33 Using the similarity of equations for Beam StaticsBeam deflection Or integrating Unit = kN·m 2 /EI Unit = kN·m 3 /EI Dr Yan Zhuge lecturer notes

Bina Nusantara University 34 Theorem 1: The slope at a point in the real beam is numerically equal to the shear at the corresponding point in the conjugate beam. Theorem 2: The displacement of a point in the real beam is numerically equal to the moment at the corresponding point in the conjugate beam. Conjugate Beam Dr Yan Zhuge lecturer notes

Bina Nusantara University 35 Conjugate Beam Draw the conjugate beam for the real beam with a proper boundary conditions Load the conjugate beam with the real beam’s M/EI diagram. This loading is directed downward when M/EI is positive and upward when M/EI is negative Determine the statics of the conjugate beam: reactions, Shear force and moments Shear force V corresponds to the slope  of the real beam, moment M corresponds to the displacement v of the real beam. Dr Yan Zhuge lecturer notes

Bina Nusantara University 36 Conjugate Beam REAL BEAM CONJUGATE BEAM Dr Yan Zhuge lecturer notes

Bina Nusantara University 37 Conjugate Beam REAL BEAM CONJUGATE BEAM Dr Yan Zhuge lecturer notes

Bina Nusantara University 38 Conjugate Beam REAL BEAM CONJUGATE BEAM Dr Yan Zhuge lecturer notes

Bina Nusantara University 39 Conjugate Beam REAL BEAM CONJUGATE BEAM Dr Yan Zhuge lecturer notes

Bina Nusantara University 40 Determine the maximum deflection of the steel beam shown in the figure. E = 200 GPa, I = 60(10 6 ) mm 4. A 9 m 8 kN B x 3 m 2 kN 6 kN Conjugate Beam Dr Yan Zhuge lecturer notes

Bina Nusantara University 41 8 kN A B x 18kNm A’ B’ x 18/EI Conjugate Beam Real Beam 45/EI 63/EI Maximum deflection occurs at the point where the slope is zero This corresponds to the same point in the conjugate beam where the shear is zero 2 kN 6 kN 9 m 3 m 81/EI 27/EI Conjugate Beam Dr Yan Zhuge lecturer notes